What a derivative is and how to find one

Updated October 20, 2026

The derivative of a function tells you how fast its output is changing at a single point: it is the slope of the tangent line to the graph there. It is written f′(x) or dy/dx. For a power of x, the power rule gives the derivative quickly: the derivative of xⁿ is nxⁿ⁻¹, so the derivative of x³ is 3x².

What a derivative means

A straight line has one slope everywhere. A curve does not: the graph of y = x² is flat at the bottom and gets steeper as you move right. The derivative is a slope that changes from point to point. At each x, it is the slope of the tangent line, the line that just touches the curve there and points the same way the curve is heading.

The same number is also a rate of change. If s(t) is a car's distance in metres after t seconds, then s′(t) is how many metres per second the distance is growing at that instant: the speed shown on the speedometer, rather than the average speed for the whole trip.

Two common notations mean the same thing. Lagrange's f′(x), read f prime of x, names the derivative of the function f. Leibniz's dy/dx, read d y d x, reminds you that it is a ratio: a tiny change in y over a tiny change in x.

The limit definition

To find the slope at one point you need two points, so start with two: x and a nearby x + h. The slope of the secant line through them is the average rate of change, (f(x + h) − f(x)) ÷ h. Now let h shrink towards 0. The second point slides along the curve towards the first, and the secant line turns into the tangent line. The value the slope approaches is the derivative.

You cannot simply put h = 0, because that divides by zero. Instead, simplify algebraically until h cancels from the denominator, and then let h go to 0. Doing this once by hand shows where the shortcut rules come from.

ProblemUse the limit definition to find the derivative of $f (x) = x^{2}$ .

$f' (x)$ $= \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
Start from the definition.
$= \lim_{h \to 0} \frac{{(x + h)}^{2} - x^{2}}{h}$
Substitute f(x + h) = (x + h)² and f(x) = x².
$= \lim_{h \to 0} \frac{x^{2} + 2 x h + h^{2} - x^{2}}{h}$
Expand (x + h)².
$= \lim_{h \to 0} \frac{2 x h + h^{2}}{h}$
The x² terms cancel.
$= \lim_{h \to 0} (2 x + h)$
Divide each term by h. Now there is no division by zero.
$= 2 x$
As h goes to 0, 2x + h goes to 2x.

Answer $f' (x) = 2 x$

The basic rules

Nobody uses the limit definition for every problem. These rules, each proved from the definition, cover most functions you meet in a first calculus course:

Constant rule: the derivative of a constant is 0, because a flat line has slope 0. The derivative of 7 is 0.
Power rule: the derivative of xⁿ is nxⁿ⁻¹. Bring the power down in front and lower it by 1. The derivative of x⁵ is 5x⁴, and of x is 1.
Constant multiple rule: a number in front stays there. The derivative of 4x³ is 4 × 3x² = 12x².
Sum and difference rule: differentiate term by term. The derivative of 3x⁴ − 5x² + 7x − 2 is 12x³ − 10x + 7.
A few functions to know: the derivative of sin x is cos x, of cos x is −sin x, of eˣ is eˣ, and of ln x is 1/x.
The power rule also works for negative and fractional powers. Rewrite 1/x as x⁻¹ and √x as x^(1/2) first: their derivatives are −x⁻² and ½x^(−1/2).

The product, quotient and chain rules

Products, quotients and functions inside other functions need their own rules. The derivative of a product is not the product of the derivatives, which is one of the most common errors in calculus.

Product rule: (uv)′ = u′v + uv′. For y = x² sin x, take u = x² and v = sin x, so y′ = 2x sin x + x² cos x.

Quotient rule: (u/v)′ = (u′v − uv′) ÷ v². A way to remember the order is low d high minus high d low, over low squared. For y = (x + 1)/(x − 1), y′ = ((1)(x − 1) − (x + 1)(1)) ÷ (x − 1)² = −2/(x − 1)².

Chain rule: for a function inside another, differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. For y = (3x² + 1)⁵, the outside is a fifth power and the inside is 3x² + 1, so y′ = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴. The example below uses all three ideas at once.

ProblemDifferentiate $f (x) = x^{3} {(2 x + 1)}^{4}$ .

$u$ $= x^{3},$ $v$ $= {(2 x + 1)}^{4}$
It is a product, so name the two factors for the product rule.
$u'$ $= 3 x^{2}$
Power rule.
$v'$ $= 4 {(2 x + 1)}^{3} \cdot 2$ $= 8 {(2 x + 1)}^{3}$
Chain rule: differentiate the fourth power, keep 2x + 1 inside, then multiply by its derivative, 2.
$f' (x)$ $= 3 x^{2} {(2 x + 1)}^{4} + x^{3} \cdot 8 {(2 x + 1)}^{3}$
Product rule: u′v + uv′.
$f' (x)$ $= x^{2} {(2 x + 1)}^{3} (3 (2 x + 1) + 8 x)$
Factor out the common x²(2x + 1)³.
$f' (x)$ $= x^{2} {(2 x + 1)}^{3} (14 x + 3)$
Inside the bracket, 6x + 3 + 8x = 14x + 3.

Answer $x^{2} {(2 x + 1)}^{3} (14 x + 3)$

Worked example: the equation of a tangent line

A tangent line needs a point and a slope. The function gives the point, and the derivative gives the slope at that point. Then point slope form, y − y₁ = m(x − x₁), does the rest.

ProblemFind the tangent line to $f (x) = x^{3} - 2 x$ at $x = 2$ .

$f (2)$ $= 8 - 4$ $= 4$
The point of contact is (2, 4).
$f' (x)$ $= 3 x^{2} - 2$
Differentiate with the power rule, term by term.
$f' (2)$ $= 3 (4) - 2$ $= 10$
The slope of the tangent at x = 2 is 10.
$y - 4$ $= 10 (x - 2)$
Point slope form with the point (2, 4) and slope 10.
$y$ $= 10 x - 16$
Distribute and add 4. Check: 10(2) − 16 = 4, so the line passes through (2, 4).

Answer $y = 10 x - 16$

Derivatives as rates in real life

Suppose a ball is thrown upward and its height in feet after t seconds is h(t) = −16t² + 64t. Its velocity is the derivative, h′(t) = −32t + 64 feet per second. At t = 1 the ball is rising at 32 ft/s. At t = 2 the velocity is 0, so the ball is at the top of its flight; after that the velocity is negative and the ball falls.

That last idea is one of the most useful in calculus: where a smooth function reaches a peak or a valley, its tangent line is flat and its derivative is 0. Setting a derivative equal to 0 is how you find the largest area, the lowest cost or the highest point. The units of a derivative are always output units per input unit, such as feet per second or dollars per item.

Common mistakes

Most derivative errors are not about calculus but about algebra and rules applied too quickly:

Multiplying derivatives for a product. The derivative of x² sin x is not 2x cos x; use the product rule.
Forgetting the chain rule's last step. The derivative of (3x + 1)⁵ is 5(3x + 1)⁴ × 3, not just 5(3x + 1)⁴.
Mixing up the order in the quotient rule, which flips the sign of the answer.
Using the power rule on an exponential. The derivative of 2ˣ is not x × 2ˣ⁻¹; the variable is in the exponent, so a different rule applies.
Forgetting that a constant has derivative 0, or that a constant multiple stays.
Using f′(x) as the y-value of a tangent line. The derivative is the slope; the point comes from f(x).

How to practise and check your answers

Practise the rules one at a time, then mix them, because recognising which rule a problem needs is the real skill. Before differentiating, rewrite roots and fractions as powers. After differentiating, simplify and factor.

To check an answer, graph the function and your derivative on a graphing calculator or free graphing website. Wherever the function is rising, your derivative should be positive; where it is falling, negative; at each peak or valley, zero. You can also estimate the slope near a point numerically: for f(x) = x² at x = 3, the secant slope from 3 to 3.001 is 6.001, close to f′(3) = 6.

Questions

What is a derivative in simple terms?

A derivative tells you how fast something is changing at one instant. On a graph, it is the slope of the curve at a single point.

What is the difference between a derivative and slope?

Slope is a single number for a straight line. A derivative gives the slope of a curve at each point, so it is usually a function of x rather than one number.

What is the power rule?

The derivative of xⁿ is nxⁿ⁻¹: multiply by the exponent, then subtract 1 from the exponent. For example, the derivative of x⁴ is 4x³.

When does a function not have a derivative?

A function has no derivative where its graph has a sharp corner, a break or jump, or a vertical tangent. For example, y = |x| has a corner at x = 0, so it has no derivative there.

When do students learn derivatives?

Derivatives are the first big topic of calculus, usually taken in the last year or two of high school, such as AP Calculus, or in a first college math course. Solid algebra, functions and slope come first.